# Full text of "Treatise On Analysis Vol-Ii"

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```16    BAIRE'S THEOREM AND ITS CONSEQUENCES        97

28. Let /be an indefinitely different iable real-valued function defined on an open interval
]a, b[ of R. Suppose that for each point x e ]a, b[ there exists an integer N(JC) such that
~D™(x)f(x) = 0. Show that /is a polynomial. (One may proceed as follows:

(a)    Let G be the open set of points jc e ]a, b[ such that in some neighborhood of x
the function / coincides with a polynomial, and let F be the complement of G in
]a, b[. Show that F has no isolated points (3.10.10).

(b)    For each integer «, let EB be the closed subset of F consisting of the points x e F
such that Dn/(X) = 0. If F is not empty, show that there exists a nonempty open
interval I c ]a, b[ and an integer N such that F n I is nonempty and is contained in
EN (use Baire's theorem). Then deduce from (a) that F n I <= En for all n > N.

(c)    Deduce from (b) that F n I is nowhere dense in I, and then that DN/(JC) = 0 on
every component interval of G n I. Hence derive a contradiction of the hypothesis

29.    Let E, F be two separable Banach spaces, F' the dual of F. Suppose that F' is contained
in a Hausdorff locally convex space G, and that the topology on F' induced by that
of G is coarser than the weak topology of F'. Let u : E -> G be a continuous linear
mapping.

(a)    Show that, for every ball B in the Banach space F', the inverse image w~x(B)
is a closed subset of E. (Use (12.15.9) and (12.15.8.1).)

(b)    Suppose that there exists a nonmeager subset A of E such that u(x) e F' for
all x e A. Show that u(E) c F' and that u is continuous for the topology defined by
the norm on F'.

30.    Let E be a Fr6chet space, and let (wa) be a family of continuous linear mappings of E
into a normed space F. Suppose that there exists a nonmeager subset A of E such that
for each x e A the set of points ua(x) is bounded in F. Show that the family (w«) is
equicontinuous. (For each integer n g: 1 , consider the set of points x e E such that

ll««(*)ll ~ n f°r eacn index a.)ecessary and sufficient condition for the equation
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